Probability and Stochastic Modeling

"Preface: This book is intended as a first course in probability with an emphasis on stochastic modeling. Distinctive features of the book concern its contents and format as well. The Contents. The exposition is rigorous; with rare exceptions, all assertions are proven; almost every topic found in a traditional introductory probability course is covered. On the other hand, the book pays substantial attention to stochastic modeling, which is atypical for first-course textbooks on Probability. Cases in point are Markov chains, birth-death processes (including queuing processes), reliability models, and other topics, both theoretical and applied. We also consider a number of concrete models (for example, a model of financial markets, or the principal components scheme); sometimes, even with real world data. The goal here is not to teach particular models or numerical methods but rather help the student to better appreciate general concepts and theoretical results, and demonstrate practical possibilities andrestrictions of different approaches under consideration. The same concerns examples and exercises with use of Excel. Besides the traditional material, we also pay attention to topics usually skipped (or almost skipped) in introductory courses, though inthe author's opinion, they are becoming increasingly important. In particular, this concerns martingales, classification of dependency structures, and risk evaluation. The format of the book. The material is presented in the form of two nested "routes". Route 1 contains the basic material designed for a one semester course. This material is self-contained and has a moderate level of difficulty. Route 2 contains all of Route 1, offers a more complete exposition, and is suited for a two-semester course or self-study"--

Vladimir I. Rotar is a professor in the Department of Mathematics and Statistics at San Diego State University. Dr. Rotar has authored four books and more than 100 scientific papers on probability theory and its applications in leading mathematical journals.

Probability Space: Sample Space, Events, Probability. Independence and Conditional Probabilities. Discrete Random Variables. Generating Functions. The Extinction Problem. Random Walk Revisited. More about Markov Chains. Continuous r.v.'s. Distributions and Expectations in the General Case. Simulation and the Monte Carlo Method. Moment Generating Functions. The Bernstein-Chernoff Bounds. Covariance Analysis and Multivariate Normal Distribution. Some Markov Stochastic Processes and Stochastic Dynamic Models of Real Phenomena. Martingales. Some Other Types of Dependencies. Dependencies Not Connected with Evolution in Time. The Central Limit Theorem. Reliability Models and Renewal Processes. Risk Assessment and Comparison of r.v.'s. Brownian Motion. Almost Sure Behavior of Sums of r.v.'s. Appendix. References. Answers to Exercises. Index.